#8922626

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3921 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,5)(3,7,9,11)(4,8,12,13)(10,14,15,16)(17,18)(19,27)(20,23)(21,25)(22,30)(24,28)(26,32)(29,31)(33,34)(35,42)(36,41)(37,40)(38,39)(43,48)(44,47)(45,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)(17,19,23,28)(18,24,20,27)(21,30,31,26)(22,25,32,29)(35,37)(40,42)(43,44)(47,48);(3,9)(4,12)(7,11)(8,13)(18,20)(22,32)(24,27)(25,29)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(1,4,6,12)(2,8,5,13)(3,10,9,15)(7,14,11,16)(17,20,23,18)(19,27,28,24)(21,25,31,29)(22,30,32,26)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,5,6,2)(3,11,9,7)(4,8,12,13)(10,14,15,16)(17,21,23,31)(18,25,20,29)(19,26,28,30)(22,27,32,24)(33,37,39,35)(34,40,38,42)(36,43,46,44)(41,48,45,47);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,15)(13,16)(17,22,23,32)(18,26,20,30)(19,25,28,29)(21,27,31,24)(33,38,39,34)(35,40,37,42)(36,45,46,41)(43,47,44,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=gi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8829822);